2 added cptness assumption, corrected example at end

Chern-Weil theory tells us that the integral Chern classes of a flat bundle over a compact manifold (i.e. a bundle admitting a flat connection) are all torsion. Given a compact manifold $M$ whose integral cohomology contains torsion, one can then ask which (even-dimensional) torsion classes appear as the Chern classes of flat bundles. What is known about this question? I would be interested both in statements about specific manifolds and about general (non)-realizability results.

One specific thing that I know: if $S$ is a non-orientable surface, then there is a flat bundle $E\to S$ whose first Chern class is the generator of $H^2 (S; \mathbb{Z}) = \mathbb{Z}/2$. This shows up, for example, in papers of C.-C. Melissa Liu and Nan-Kuo Ho.

One specific thing I don't know: if $S_1$ and $S_2$ are non-orientable surfacesAs Johannes pointed out in the comments, can this also shows that the generator fundamental class of $H^4 (S_1 \times S_2; Z) = \mathbb{Z}/2$ a product of surfaces can be realized by a flat bundle.

However, I suspect that for a product of 3 Klein bottles, not all the 4-dimensional torsion classes can be realized as second Chern class classes of a flat bundle? bundles. In fact, I strongly suspect think I know a proof of this if one restricts to unitary flat connections: the answer is no..space of unitary representations has too few connected components.

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# Which torsion classes in integral cohomology are Chern classes of flat bundles?

Chern-Weil theory tells us that the integral Chern classes of a flat bundle (i.e. a bundle admitting a flat connection) are all torsion. Given a manifold $M$ whose integral cohomology contains torsion, one can then ask which (even-dimensional) torsion classes appear as the Chern classes of flat bundles. What is known about this question? I would be interested both in statements about specific manifolds and about general (non)-realizability results.

One specific thing that I know: if $S$ is a non-orientable surface, then there is a flat bundle $E\to S$ whose first Chern class is the generator of $H^2 (S; \mathbb{Z}) = \mathbb{Z}/2$. This shows up, for example, in papers of C.-C. Melissa Liu and Nan-Kuo Ho.

One specific thing I don't know: if $S_1$ and $S_2$ are non-orientable surfaces, can the generator of $H^4 (S_1 \times S_2; Z) = \mathbb{Z}/2$ be the second Chern class of a flat bundle? I strongly suspect the answer is no...